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Question Number 66815 by Rio Michael last updated on 20/Aug/19

solve the congruence equation 6x ≡ 4 (mod 5) i need help please with some explanations

$${solve}\:{the}\:{congruence}\:{equation}\: \\ $$$$\:\:\mathrm{6}{x}\:\equiv\:\mathrm{4}\:\left({mod}\:\mathrm{5}\right)\:\:{i}\:{need}\:{help}\:{please}\:{with}\:{some}\:{explanations} \\ $$

Commented by mathmax by abdo last updated on 20/Aug/19

if we consider congruence modulo 5 we have 6x ≡4 [5] ⇒6^− x^− =4^− but 6^− =5^− +1^− =1^− and 4^− =(−1)^− ⇒ x^− =−1[5] ⇒x+1 =5k ⇒x =5k−1 with k ∈Z .

$${if}\:{we}\:{consider}\:{congruence}\:{modulo}\:\mathrm{5}\:\:{we}\:{have}\: \\ $$$$\mathrm{6}{x}\:\equiv\mathrm{4}\:\left[\mathrm{5}\right]\:\Rightarrow\overset{−} {\mathrm{6}}\overset{−} {{x}}=\overset{−} {\mathrm{4}}\:\:\:\:{but}\:\overset{−} {\mathrm{6}}=\overset{−} {\mathrm{5}}\:+\overset{−} {\mathrm{1}}=\overset{−} {\mathrm{1}}\:{and}\:\overset{−} {\mathrm{4}}=\left(−\mathrm{1}\overset{−} {\right)}\:\Rightarrow \\ $$$$\overset{−} {{x}}=−\mathrm{1}\left[\mathrm{5}\right]\:\Rightarrow{x}+\mathrm{1}\:=\mathrm{5}{k}\:\:\:\Rightarrow{x}\:=\mathrm{5}{k}−\mathrm{1}\:\:{with}\:{k}\:\in{Z}\:. \\ $$$$ \\ $$

Answered by Rasheed.Sindhi last updated on 20/Aug/19

6x≡4+4×5=24(mod 5) x≡4(mod 5)

$$\mathrm{6}{x}\equiv\mathrm{4}+\mathrm{4}×\mathrm{5}=\mathrm{24}\left({mod}\:\mathrm{5}\right) \\ $$$$\:\:{x}\equiv\mathrm{4}\left({mod}\:\mathrm{5}\right) \\ $$

Answered by mr W last updated on 20/Aug/19

6x=5n+4=5(n−4)+24 x=((5(n−4))/6)+4 with n−4=6m ⇒x=5m+4 i.e. x≡4(mod 5)

$$\mathrm{6}{x}=\mathrm{5}{n}+\mathrm{4}=\mathrm{5}\left({n}−\mathrm{4}\right)+\mathrm{24} \\ $$$${x}=\frac{\mathrm{5}\left({n}−\mathrm{4}\right)}{\mathrm{6}}+\mathrm{4} \\ $$$${with}\:{n}−\mathrm{4}=\mathrm{6}{m} \\ $$$$\Rightarrow{x}=\mathrm{5}{m}+\mathrm{4} \\ $$$${i}.{e}.\:{x}\equiv\mathrm{4}\left({mod}\:\mathrm{5}\right) \\ $$

Commented by Rio Michael last updated on 20/Aug/19

i still don′t undestand how these equations are solved

$${i}\:{still}\:{don}'{t}\:{undestand}\:{how}\:{these}\:{equations}\:{are}\:{solved} \\ $$

Commented by mr W last updated on 21/Aug/19

don′t you understand what 6x≡4 (mod 5) means or where is your problem?

$${don}'{t}\:{you}\:{understand}\:{what} \\ $$$$\mathrm{6}{x}\equiv\mathrm{4}\:\left({mod}\:\mathrm{5}\right)\:{means}\:{or}\:{where}\:{is}\:{your} \\ $$$${problem}? \\ $$

Commented by Rio Michael last updated on 21/Aug/19

i think so sir,because am new to this further mathematics stuffs

$${i}\:{think}\:{so}\:{sir},{because}\:{am}\:{new}\:{to}\:{this}\:{further}\:{mathematics}\:{stuffs} \\ $$

Commented by mr W last updated on 21/Aug/19

a≡b (mod c) means when a and b are divided by c, they have the same remainder. in other word: a=nc+b

$${a}\equiv{b}\:\left({mod}\:{c}\right)\:{means} \\ $$$${when}\:{a}\:{and}\:{b}\:{are}\:{divided}\:{by}\:{c},\:{they} \\ $$$${have}\:{the}\:{same}\:{remainder}. \\ $$$${in}\:{other}\:{word}: \\ $$$${a}={nc}+{b} \\ $$

Commented by Rio Michael last updated on 25/Aug/19

thank you so much sir

$${thank}\:{you}\:{so}\:{much}\:{sir} \\ $$

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